Preheat the oven to 425°. In a large ovenproof skillet, heat the olive oil. Season the chicken pieces with salt and pepper. Add them to the skillet skin side down and cook over moderately high heat, turning once, until browned, about 12 minutes; transfer the chicken to a plate. Discard the fat.
Add the garlic and shallot to the skillet and cook until fragrant, about 30 seconds. Return the thighs and drumsticks to the skillet, add the chicken stock and bring to a boil. Transfer the skillet to the oven and braise for 20 minutes. Add the cream and the breasts and wings and braise for 15 minutes longer. Transfer the chicken to a platter, cover with foil and keep warm.
Set the skillet over high heat and boil until the sauce is reduced to 1 cup, about 7 minutes. Scrape it into a blender or food processor, add the arugula and puree until smooth. If desired, add the butter and puree until blended. Season the sauce with salt and pepper, pour it over the chicken and serve.
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